Proof of CRT Coverage Maximality for up to Seven Distinct Odd Prime Factors

This research memorandum establishes a complete mathematical proof for the CRT Coverage Maximality inequality for integers possessing at most seven distinct odd prime factors (\(k ≤ 7\)). Given a product of powers of distinct odd primes, the inequality compares the exact size of the union of arbitrarily chosen residue classes modulo each divisor greater than 1 with an alternating sum benchmark evaluated over pairwise-coprime families of those divisors. Methodology & Key ResultsThe proof avoids floating-point computations or numerical searches, relying entirely on exact rational arithmetic. The global problem is successfully reduced to prime supports via two main components: Support-Polynomial Induction: The memorandum groups an independence polynomial by prime support and proves that the full-family polynomial \(fₘ\) is coordinatewise strictly increasing on its domain. For \(k ≤ 7\), a majorization argument over the six smallest odd primes (3, 5, 7, 11, 13, 17) yields an exact uniform monotonicity margin of \(517/1280>0\). Dependency-Graph Machinery: The framework models the system as a dependency graph where vertices represent divisors and edges represent non-coprimality. Because the underlying Chinese Remainder Theorem (CRT) coordinates are independent, the structural requirements are met to apply Shearer’s dependency-graph theorem to conclude the proof. Scope & Open BoundariesThe theorem holds unconditionally for any selection of positive prime exponents and completely decoupled, arbitrary residue choices. As a point of interest for future work, the memorandum calculates the next structural coefficient, showing it becomes negative (\(c_8 = -50\)). This means the direct monotonicity certificate used here cannot be mechanically copied to the next dimension, leaving the unrestricted conjecture open for numbers with eight or more distinct odd prime factors (\(k ≥ 8\)).